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In this work, a coupled 3D thermo-elastic shell model is presented. The primary variables are the scalar sovra-temperature and the displacement vector. This model allows for the thermal stress analysis of one-layered and sandwich plates and shells embedding Functionally Graded Material (FGM) layers. The 3D equilibrium equations and the 3D Fourier heat conduction equation for spherical shells are put together into a set of four coupled equations. They automatically degenerate in those for simpler geometries thanks to proper considerations about the radii of curvature and the use of orthogonal mixed curvilinear coordinates α, β, and z. The obtained partial differential governing the equations along the thickness direction are solved using the exponential matrix method. The closed form solution is possible assuming simply supported boundary conditions and proper harmonic forms for all the unknowns. The sovra-temperature amplitudes are directly imposed at the outer surfaces for each geometry in steady-state conditions. The effects of the thermal environment are related to the sovra-temperature profiles through the thickness. The static responses are evaluated in terms of displacements and stresses. After a proper and global preliminary validation, new cases are presented for different thickness ratios, geometries, and temperature values at the external surfaces. The considered FGM is metallic at the bottom and ceramic at the top. This FGM layer can be embedded in a sandwich configuration or in a one-layered configuration. This new fully coupled thermo-elastic model provides results that are coincident with the results proposed by the uncoupled thermo-elastic model that separately solves the 3D Fourier heat conduction equation. The differences are always less than 0.5% for each investigated displacement, temperature, and stress component. The differences between the present 3D full coupled model and the the advantages of this new model are clearly shown. Both the thickness layer and material layer effects are directly included in all the conducted coupled thermal stress analyses.

This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model. This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic, composite and sandwich structures. Cylindrical and spherical panels, cylinders and plates are analyzed in orthogonal mixed curved reference coordinates. The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels. The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate. The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature. The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses. As a consequence, the sovra-temperature description is assumed to be an unknown variable of the model and it is calculated in the same way as the three displacements. The final system is based on a set of coupled homogeneous differential relations of second order in the thickness coordinate. This system is reduced in a first order differential relation system by redoubling the number of unknowns. Therefore, the exponential matrix methodology is applied to calculate the solution. The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components. After an appropriate preliminary validation, new benchmarks are discussed for several thickness ratios, geometrical data, lamination sequences, materials and sovra-temperature values imposed at the outer faces. Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation. Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis. KEYWORDSThree-dimensional exact thermo-elastic shell model; full coupled thermoelasticity; three-dimensional fourier heat conduction relation; plate and shell geometries; laminated and sandwich configurations IntroductionThe monitoring of temperature gradients in the aerospace structures is one of the main topics in the stress analysis. The study of temperature contribution on strains and stresses is very important

In this work, a 3D fully coupled hygro-elastic model is proposed. The moisture content profile is a primary variable of the model’s displacements. This generic fully coupled 3D exact shell model allows the investigations into the consequences arising from moisture content and elastic fields in terms of stresses and deformations on different plate and shell configurations embedded in composite and laminated layers. Cylinders, plates, cylindrical and spherical shells are analyzed in the orthogonal mixed curvilinear reference system. The 3D equilibrium equations and the 3D Fick diffusion equation for spherical shells are fully coupled in a dedicated system. The main advantage of the orthogonal mixed curvilinear coordinates is related to the degeneration of the equations for spherical shells to simpler geometries thanks to basic considerations of the radii of curvature. The exponential matrix method is used to solve this fully coupled model based on partial differential equations in the thickness direction. The closed-form solution is related to simply supported sides and harmonic forms for displacements and the moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. A first-order differential equation system is obtained by redoubling the number of variables. The moisture field implications are evaluated for the static analysis of the plates and shells in terms of displacement and stress components. After preliminary validations, new benchmarks are proposed for several thickness ratios, geometrical and material data, lamination sequences and moisture values imposed at the external surfaces. In the proposed results, there is clearly accordance between the uncoupled hygro-elastic model (where the 3D Fick diffusion law is separately solved) and this new fully coupled hygro-elastic model: the differences between the investigated variables (displacements, moisture contents, stresses and strains) are always less than 0.3%. The main advantages of the 3D coupled hygro-elastic model are a more compact mathematical formulation and lower computational costs. Both effects connected with the thickness layer and the embedded materials are included in the conducted hygro-elastic analyses.

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